Numerical analysis of the planewave discretization of some orbital-free and Kohn-Sham models

DOI:10.1051/m2an/2011038 www.esaim-m2an.org

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NUMERICAL ANALYSIS OF THE PLANEWAVE DISCRETIZATION OF SOME ORBITAL-FREE AND KOHN-SHAM MODELS

Eric Canc` es

, Rachida Chakir

and Yvon Maday

Abstract. In this article, we provide a priori error estimates for the spectral and pseudospectral Fourier (also called planewave) discretizations of the periodic Thomas-Fermi-von Weizs¨acker (TFW) model and for the spectral discretization of the periodic Kohn-Sham model, within the local density approximation (LDA). These models allow to compute approximations of the electronic ground state energy and density of molecular systems in the condensed phase. The TFW model is strictly convex with respect to the electronic density, and allows for a comprehensive analysis. This is not the case for the Kohn-Sham LDA model, for which the uniqueness of the ground state electronic density is not guaranteed. We prove that, for any local minimizer Φ⁰ of the Kohn-Sham LDA model, and under a coercivity assumption ensuring the local uniqueness of this minimizer up to unitary transform, the discretized Kohn-Sham LDA problem has a minimizer in the vicinity of Φ⁰ for large enough energy cut-offs, and that this minimizer is unique up to unitary transform. We then derive optimal a priori error estimates for the spectral discretization method.

Mathematics Subject Classification. 65N25, 65N35, 65T99, 35P30, 35Q40, 81Q05.

Received April 8, 2010. Revised March 22, 2011.

Published online October 24, 2011.

1. Introduction

First-principle molecular simulation allows to better understand, or to predict, the properties of matter from the fundamental laws of quantum mechanics. It is a major tool in chemistry, condensed matter physics, and materials science, used on a daily basis by hundreds of research groups in academy and industry. It is also becoming a fruitful approach in molecular biology and nanotechnologies.

In this approach, matter is described as an assembly of nuclei and electrons. At this scale, the equation that rules the interactions between these constitutive elements is the N-body Schr¨odinger equation. It can be considered (except in few special cases notably those involving relativistic phenomena or nuclear reactions) as a universal model for at least three reasons. First, it contains all the physical information of the system under consideration, so that any of the properties of this system can be deduced in theory from the Schr¨odinger

Keywords and phrases. Electronic structure calculation, density functional theory, Thomas-Fermi-von Weizs¨acker model, Kohn-Sham model, nonlinear eigenvalue problem, spectral methods.

1Universit´e Paris-Est, CERMICS, Project-team Micmac, INRIA- ´Ecole des Ponts, 6 & 8 avenue Blaise Pascal, 77455 Marne-la- Vall´ee Cedex 2, France. cances@cermics.enpc.fr

2 UPMC Univ. Paris 06, UMR 7598 LJLL, 75005 Paris, France

3 CNRS, UMR 7598 LJLL, 75005 Paris, France

4 Division of Applied Mathematics, 182 George Street, Brown University, Providence, RI 02912, USA

Article published by EDP Sciences c EDP Sciences, SMAI 2011

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equation associated to it. Second, the Schr¨odinger equation does not involve any empirical parameter, except some fundamental constants of physics (the Planck constant, the mass and charge of the electron, . . . ); it can thus be written for any kind of molecular system provided its chemical composition, in terms of natures of nuclei and number of electrons, is known. Third, this model enjoys remarkable predictive capabilities, as confirmed by comparisons with a large amount of experimental data of various types.

Of course, theN-body time-dependent Schr¨odinger equation, which is an evolution partial differential equa- tion in space dimension 3N, whereN is the number of particles (nuclei and electrons) in the system, cannot be solved by brute force numerical methods when N exceeds three or four. In order to deal with larger systems, approximations have to be resorted to. The Born-Oppenheimer approximation [4], based on the fact that nuclei are thousands of times heavier than electrons, allows to decouple the nuclear and electronic dynamics, and to consider that, in most cases, nuclei behave as point-like classical particles, and electrons are in their ground state. Several methods for computing approximations of electronic ground states have then been proposed, which can be classified in three groups:

• wavefunction methods, among which the Hartree-Fock and multiconfiguration self-consistent-field (MC- SCF) models (see [8] for a mathematical introduction);

• methods issued from the density functional theory (DFT);

• quantum Monte Carlo methods [24,25].

DFT currently is the most popular approach for it offers the best compromize between accuracy and compu- tation cost. The models originating from DFT can be classified into two categories: the orbital-free models and the Kohn-Sham models. The Thomas-Fermi-von Weizs¨acker (TFW) model falls into the first category. It is not very much used in practice, but is interesting from both a mathematical viewpoint [2,12,27] and a numerical viewpoint [35]. It indeed serves as a toy model for the analysis of the more complex electronic structure models routinely used by physicists and chemists, as well as for the development of new numerical methods [17,31]. At the other extremity of the spectrum, the Kohn-Sham models [15,21] are among the most widely used models in physics and chemistry, but are much more difficult to deal with.

Throughout this article, we adopt the system of atomic units for which = 1 (reduced Planck constant), me= 1 (mass of the electron),e= 1 (elementary charge), 4π₀= 1 (₀ being the dielectric permittivity of the vacuum). In this system of units, the charge of the electron is−1 and the charges of nuclei are positive integers.

Let us first consider an isolated molecular systemin vacuo, consisting of M nuclei of charges (z₁, . . . , zM)∈ (N\ {0})^M located at the positions (R₁, . . . , RM) ∈ (R³)^M of the physical space, and of N electrons. The electrostatic potential generated by the nuclei and felt by the electrons is

V^nuc(x) =− M k=1

zk

|x−Rk|· (1.1)

In the TFW model, as well as in any orbital-free model, the ground state electronic density of the system is obtained by minimizing anexplicit functional of the density. For the system under consideration, this model reads [27]

inf

E₀^TFW(ρ), ρ≥0, √

ρ∈H¹(R³),

R³ρ=N

, (1.2)

whereE₀^TFW(ρ) is the TFW energy functional defined as E₀^TFW(ρ) :=C_W

2

R³|∇√ρ|²+C_TF

R³ρ^5/3+

R³ρV^nuc+1

2D(ρ, ρ), and where the bilinear form D(·,·) is the Coulomb energy functionalin vacuo:

D(ρ, ρ) :=

R³

R³

ρ(x)ρ(y)

|x−y| dxdy= 4π

R³|k|⁻²ρ(k) ^∗ρ(k) dk, (1.3)

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fdenoting the Fourier transform off (normalized in such a way that the Fourier transform is the isometry of L²(R³)). Here and in the sequela^∗denotes the complex conjugate of the complex numbera. The first two terms of the TFW energy functional model the kinetic energy of the electrons;C_W is a positive real number (C_W = 1, 1/5 or 1/9 depending on the context [15]) and C_TF is the Thomas-Fermi constant (C_TF = ¹⁰₃(3π²)^2/3). The third and fourth terms respectively model the interactions between nuclei and electrons, and the interactions between electrons.

In the Kohn-Sham model, the electronic state of the closed-shell system with an even numberN = 2N of electrons is described byN Kohn-Sham orbitals Φ = (φ₁, . . . , φN)^T ∈(H¹(R³))^N satisfying the orthonormality

conditions

R³φiφj=δij, and the associated electronic density

ρ_Φ(x) := 2 N i=1

|φi(x)|².

The factor 2 in the above expression accounts for the spin. In closed-shell systems, each Kohn-Sham orbital is indeed occupied by two electrons, one with spin up and one with spin down. The Kohn-Sham ground state is obtained by solving the minimization problem

inf

E₀^KS(Φ), Φ = (φ₁, . . . , φN)^T ∈(H¹(R³))^N,

R³φiφj =δij

, (1.4)

where the Kohn-Sham energy functional reads E₀^KS(Φ) :=

N i=1

R³|∇φi|²+

R³V^nucρ_Φ+1

2D(ρ_Φ, ρ_Φ) +E_xc(ρ_Φ). (1.5) The first term models the kinetic energy, the second term the interactions between nuclei and electrons, and the third term the interaction between electrons. The fourth term, called the exchange-correlation functional, is a correction term, which is essential to describe quantitatively, and sometimes even qualitatively, the physics and chemistry of the system. The exchange-correlation functional collects the errors made in the approximations of the kinetic energy and of the interactions between electrons by respectively the first and third terms of the Kohn-Sham functional. It follows from the Hohenberg-Kohn theorem [20,26,28,33], that there exists anexact exchange-correlation functional, that is a functional of the electronic densityρfor which solving (1.4) provides the ground state electronic energy and density of theN-body electronic Schr¨odinger equation. Unfortunately, no mathematical expression of the exchange-correlation functional amenable to numerical simulations is known. It therefore has to be approximated in practice. The local density approximation (LDA) consists in approximating the exchange-correlation functional by

R³e^LDA_xc (ρ(x)) dx

wheree^LDA_xc (ρ) is an approximation of the exchange-correlation energy per unit volume in a uniform electron gas with densityρ. The resulting Kohn-Sham LDA model is well understood from a mathematical viewpoint [1,23].

On the other hand, the existence of minimizers for Kohn-Sham models based on more refined approximations of the exchange-correlation functional, such as generalized gradient approximations [1] or exact local exchange potentials [10], still is an open problem.

In the sequel, we will focus on the periodic versions of the TFW and Kohn-Sham LDA models. In the periodic setting, the simulation domain, sometimes referred to as the supercell, is no longer the whole space R³, as in (1.2)–(1.4); it is the unit cell Γ of some periodic lattice ofR³. In the periodic TFW model, periodic boundary conditions (PBC) are imposed to the density; in the periodic Kohn-Sham framework, they are imposed to the Kohn-Sham orbitals (Born-von Karman PBC). Imposing PBC at the boundary of the simulation cell is

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the standard method to compute condensed phase properties with a limited number of atoms in the simulation cell, hence at a moderate computational cost.

In most applications, the periodic TFW and Kohn-Sham models are discretized in Fourier modes, more commonly referred to as planewave basis sets in the physics and chemistry literature. This is the reason why we focus on this particular discretization method in the present work. The TFW ground state density corresponding to the nuclear potential (1.1) is known to have cusps at the nuclear positions Rk. These singularities reduces the efficiency of the planewave discretization method. In practice, the singular nuclear potentialV^nuc defined by (1.1) is usually replaced with a smoother potentialV^ion; this amounts to replacing point nuclei with smeared nuclei. We will see in Section3that, not surprisingly, the smoother the potential, the faster the convergence of the planewave approximation to the exact solution of (1.2). In the Kohn-Sham setting, this issue is addressed in a more refined way [32], for it is also considered that core electrons are not affected by the chemical environment.

The Kohn-Sham orbitals of the core electrons surrounding each nucleus (for instance the two 1selectrons of the nuclei of the second row of the periodic table) are frozen to their ground states in the isolated atom. The Kohn- Sham orbitals of the valence electrons (i.e. of the electrons which are not core electrons and are therefore affected by the chemical environment) are replaced with pseudo-orbitals, which coincide with the valence Kohn-Sham orbitals out of a so-called core region surrounding each nucleus, and are smoother than the valence Kohn-Sham orbitals inside the core region. The resulting model is similar to (1.4), but presents some differences: (i)N now denotes the number of valence electron pairs, (ii) Φ now denotes the set of the pseudo-orbitals of the valence electrons, and (iii) the nuclear potentialV^nucis replaced by apseudopotentialmodeling the Coulomb interaction between the valence electrons on the one hand, and the nuclei and the core electrons on the other hand. The pseudopotential consists of two terms: a local componentV_local(whose associated operator is the multiplication by the functionV_local) and a nonlocal component (an operator whose expression is given in Section 4). As a consequence, the second term in the Kohn-Sham energy functional (1.5) is replaced by

Γ

ρ_ΦV_local+ 2 N i=1

φi|Vnl|φi.

The pseudopotential approximation has two main advantages: first, it allows to deal with heavy nuclei (for which core electrons are relativistic) within a non-relativistic framework, and, second, it reduces the computational cost by reducing the numberN of orbitals to be computed and by regularizing these orbitals (hence increasing the rate of convergence of the planewave approximation). The pseudopotential appoximation gives satisfactory results in most cases, but sometimes fails. A mathematical analysis of the pseudopotential approximation is still lacking.

The purpose of this article is to provide an analysis of the Fourier spectral and pseudospectral discretizations of the periodic TFW and Kohn-Sham LDA models, following our first contribution [6] dealing with simpler non- linear eigenvalue problems. As far as we know, our results are the first ones presenting the optimal convergence rate for the ground state energy and eigenpairs, both for the TFW type problems where some papers already existed, and for the Kohn-Sham problem where no numerical analysis in terms of convergence was available.

Previous contributions in the numerical analysis of electronic structure models are actually very few. In [35], the convergence of the ground state energy and eigenpair is established for the Galerkin discretization of a convex TFW model, but no optimal rate of convergence is provided. In [22], the authors have considered the Thomas-Fermi-Dirac-von Weizs¨acker model, that is a non convex model entering in the category of orbital-free DFT models. Under an hypothesis of ellipticity of the second order derivative of the Lagrangian associated with the minimization problem, they prove that the discrete problem, based on a P1-Lagrange finite element approximation has, locally, a unique discrete solution that converges at the optimal rate in the energy norm.

The convergence of the eigenvalue is also obtained, but is not optimal.

More recently, Zhouet al. have analyzed a non-convex orbital-free model [14]; again with an assumption of local inversibility in the vicinity of the ground states, the convergence of the minimizers of the discrete problem to the set of the minimizers of the continuous problem is established, but no convergence rate is actually proven.

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All the results mentioned above deal witha priorianalysis. The results abouta posteriori error analysis are even more seldom. We refer to [13,29] for the available results in this direction and improvements of the basic approximation by either postprocessing, or thanks to adaptivity.

This article is organized as follows. In Section 2, we briefly introduce the functional setting used in the formulation and the analysis of the planewave discretization of periodic orbital-free and Kohn-Sham models.

In Section 3, we provide a priorierror estimates for the planewave discretization of the periodic TFW model, including numerical integration. In Section4, we deal with the periodic Kohn-Sham LDA model.

2. Basic Fourier analysis for planewave discretization methods

Throughout this article, we denote by Γ the simulation cell, by R the periodic lattice, and by R^∗ the dual lattice. For simplicity, we assume that Γ = [0, L)³ (L > 0), in which case R is the cubic lattice LZ³, and R^∗ = ²_L^πZ³. Our arguments can be easily extended to the general case. For k ∈ R^∗, we denote by ek(x) =|Γ|^−1/2e^ik·x the planewave with wavevectork. The family (ek)k∈R^∗ forms an orthonormal basis of

L²_#(Γ,C) :=

u∈L²_loc(R³,C)|uR-periodic , and for allu∈L²_#(Γ,C),

u(x) =

k∈R^∗

ukek(x) with uk= (ek, u)_L2

# =|Γ|^−1/2

Γ

u(x)e^−ik·xdx.

In our analysis, we will mainly consider real valued functions. We therefore introduce the Sobolev spaces of real valuedR-periodic functions

H_#^s(Γ) := u(x) =

k∈R^∗

ukek(x)|

k∈R^∗

(1 +|k|²)^s|uk|²<∞and∀k, u₋k=u^∗_k

,

s∈R, endowed with the inner products (u, v)H^s

# =

k∈R^∗

(1 +|k|²)^su^∗_kvk.

ForNc∈N, we denote by

VNc=

⎧⎨

⎩

k∈R^∗| |k|≤^2πLN_c

ckek| ∀k, c₋k=c^∗_k

⎫⎬

⎭ (2.1)

(the constraintsc₋k =c^∗_k imply that the functions ofVNc are real valued). The norm| · |used in the definition ofVN_c is the Euclidian norm. The plane waveek belongs toVN_c if and only if its kinetic energy ¹₂|k|²is smaller than the energy cut-offE_c = ²_L^π₂²N_c². For alls∈R, and eachv ∈H_#^s(Γ), the best approximation ofv inVNc

foranyH_#^r-norm, r≤s, is

ΠNcv=

k∈R^∗| |k|≤^2π_LN_c

vkek.

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The more regularv(the regularity being measured in terms of the Sobolev normsH^r), the faster the convergence of this truncated series tov: for all real numbersr andswithr≤s, we have for eachv∈H_#^s(Γ),

v−ΠNcvH_#^r = min

v_Nc∈V_Ncv−vNcH^r_# ≤ L

2π s−r

N_c^−(s−r)v−ΠNcvH_#^s

≤ L

2π s−r

N_c^−(s−r)vH^s

#. (2.2)

ForNg ∈N\ {0}, we denote by φ^FFT,Ng the discrete Fourier transform on the cartesian gridGNg := _N^L_gZ³ of the functionφ∈C_#⁰(Γ,C), where

C_#⁰(Γ,C) :=

u∈C⁰(R³,C)|uR-periodic . Recall that if φ =

k∈R^∗φkek ∈ C_#⁰(Γ,C), then the discrete Fourier transform of φ is the NgR^∗-periodic sequenceφ^FFT,Ng = (φ^FFT_k ^,Ng)k∈R^∗, where

φ^FFT_k ^,Ng = 1 Ng³

x∈GNg∩Γ

φ(x)e^−ik·x=|Γ|^−1/2

K∈R^∗

φk+NgK.

We now introduce the subspaces

W_N^1D_g =

Span

e^ily|l∈ 2π

LZ, |l| ≤ 2π L

Ng−1 2

(Ng odd), Span

e^ily|l∈ 2π

LZ, |l| ≤ 2π L

Ng

2

⊕C

e^iπNgy/L+ e^−iπNgy/L

(Ng even),

(W_N^1D_g ∈C_#^∞([0, L),C) and dim(W_N^1D_g) =Ng), andW_N^3D_g =W_N^1D_g ⊗W_N^1D_g ⊗W_N^1D_g. Note thatW_N^3D_g is a subspace ofH_#^s(Γ,C) of dimensionN_g³, for alls∈R, and that if Ng is odd,

W_N^3D_g = Span

ek |k∈ R^∗= 2π

LZ³, |k|_∞≤ 2π L

Ng−1 2

·

It is then possible to define the interpolation projectorINg fromC_#⁰(Γ,C) ontoW_N^3D_g by [INg(φ)](x) =φ(x) for allx∈ GNg. It holds

∀φ∈C_#⁰(Γ,C),

Γ

IN_g(φ) =

x∈GNg∩Γ

L Ng

₃

φ(x). (2.3)

The coefficients of the expansion of IN_g(φ) in the canonical basis of W_N^3D_g is given by the discrete Fourier transform ofφ. In particular, whenNg is odd, we have the simple relation

INg(φ) =|Γ|^1/2

k∈R^∗| |k|∞≤^2πL

_Ng−1

2

φ^FFT_k ^,Ngek. It is easy to check that ifφis real-valued, then so isINg(φ).

We will assume in the sequel that Ng ≥4Nc+ 1. Using the properties of Gauss integration, we then have for allv₄N_c ∈V₄N_c,

Γ

v₄N_c =

x∈GNg∩Γ

L Ng

₃

v₄N_c(x) =

Γ

IN_g(v₄N_c). (2.4)

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The following lemma collects some technical results which will be useful for the numerical analysis of the planewave discretization of orbital-free and Kohn-Sham models.

Lemma 2.1. Let Nc∈N^∗ andNg∈N^∗ such thatNg≥4Nc+ 1.

(1) Let V be a function of C_#⁰(Γ,C)andvNc andwNc be two functions of VNc. Then

Γ

INg(V vNcwNc) =

Γ

INg(V)vNcwNc, (2.5)

Γ

INg(V|vNc|²)

≤ VL^∞vNc²L²

#. (2.6)

(2) Let s >3/2,0≤r≤s, andV a function ofH_#^s(Γ). Then, (1− INg)(V)

H_#^r ≤ Cr,sNg^−(s−r)VH^s

#, (2.7)

Π₂Nc(INg(V))

L²

# ≤

Γ

INg(|V|²) ₁/2

, (2.8)

Π₂Nc(INg(V))

H^s

# ≤ (1 +Cs,s)VH_#^s, (2.9)

for constantsCr,s independent ofV. Besides if there existsm >3andC∈R+such that |Vk| ≤C|k|^−m for allk∈ R^∗, then there exists a constantCV independent ofNc andNg such that

Π₂Nc(1− INg)(V)

H_#^r ≤ CVN_c^r+3/2N_g^−m. (2.10) (3) Let φ be a Borel function from R+ toR such that there exists Cφ ∈R+ for which|φ(t)| ≤Cφ(1 +t²)

for allt∈R+. Then, for allvN_c∈VN_c,

Γ

INg(φ(|vNc|²))

≤ Cφ

|Γ|+vNc⁴_L4

#

. (2.11)

Proof. For z₂N_c ∈ V₂N_c, it holds IN_g(V)z₂N_c ∈ W₂N_g−1. It therefore follows from the properties of Gauss integration that

Γ

IN_g(V z₂N_c) =

x∈GNg∩Γ

L Ng

₃

V(x)z₂N_c(x)

=

x∈GNg∩Γ

L Ng

₃

(IN_g(V))(x)z₂N_c(x)

=

Γ

INg(V)z₂Nc. (2.12)

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The functionvN_cwN_c being inV₂N_c, (2.5) is proved. Moreover, as|vN_c|²∈V₄N_c, it holds

Γ

INg(V|vNc|²) =

x∈GNg∩Γ

L Ng

₃

V(x)|vNc(x)|²

≤ VL∞

x∈GNg∩Γ

L Ng

₃

|vN_c(x)|²

= VL∞

Γ

|vN_c|².

Hence (2.6). The estimate (2.7) is proved in [11], p. 272. To prove (2.8), we notice that Π2N_c(IN_g(V))²_L2

# ≤ IN_g(V)²_L2

#

=

Γ

(IN_g(V))^∗(IN_g(V))

=

x∈GNg∩Γ

L Ng

₃

(INg(V))(x)^∗(INg(V))(x)

=

x∈GNg∩Γ

L Ng

₃

|V(x)|²

=

Γ

IN_g(|V|²).

The bound (2.9) is a straightforward consequence of (2.7):

Π₂Nc(INg(V))H^s

# ≤ INg(V)H^s

# ≤ VH^s

#+(1− INg)(V)H^s

# ≤(1 +Cs,s)VH^s

#. Now, we notice that

Π₂N_c(IN_g(V)) = |Γ|^1/2

k∈R^∗| |k|≤^4π_LNc

V_k^FFT,Ngek

=

k∈R^∗| |k|≤^4πLNc

K∈R^∗

Vk+N_gK

ek. (2.13)

From (2.13), we obtain Π₂N_c(1− IN_g)(V)²

H^s_# =

k∈R^∗| |k|≤^4πLN_c

(1 +|k|²)^s

K∈R^∗\{0}

Vk+N_gK

2

≤

⎛

⎝

k∈R^∗| |k|≤^4πLN_c

(1 +|k|²)^s

⎞

⎠ max

k∈R^∗| |k|≤^4πLN_c

K∈R^∗\{0}

Vk+N_gK

2

.

On the one hand,

k∈R^∗| |k|≤^4π_LN_c

(1 +|k|²)^s ∼

N_c→∞

32π 2s+ 3

4π L

₂s

N_c^2s+3,

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and on the other hand, we have for eachk∈ R^∗ such that|k| ≤ ⁴_L^πNc,

K∈R^∗\{0}

Vk+N_gK

≤ C

K∈R^∗\{0}

1

|k+NgK|^m

≤ C C₀ L

2π m

N_g^−m, where

C₀= max

y∈R³| |y|≤1/2

K∈Z³\{0}

1

|y−K|^m·

The estimate (2.10) then easily follows. Let us finally prove (2.11). Using (2.3) and (2.4), we have

Γ

INg(φ(|vNc|²)) =

x∈GNg∩Γ

L Ng

₃

φ(|vNc(x)|²)

≤ Cφ

x∈GNg∩Γ

L Ng

₃

(1 +|vN_c(x)|⁴)

= Cφ

Γ

(1 +|vN_c|⁴) =Cφ

|Γ|+vN_c⁴_L4

#

.

This completes the proof of Lemma 2.1.

3. Planewave approximation of the periodic TFW model

The periodic TFW problem reads as follows:

I^TFW= inf

E^TFW(ρ), ρ∈RN

, (3.1)

where

RN =

ρ≥0|√

ρ∈H_#¹(Γ),

Γ

ρ=N

is the set of admissible periodic densities, and where E^TFW(ρ) =C_W

2

Γ

|∇√ρ|²+C_TF

Γ

ρ^5/3+

Γ

ρV^ion+1

2D_Γ(ρ, ρ).

The last term of the TFW energy models the periodic Coulomb energy: forρandρ in H_#⁻¹(Γ), D_Γ(ρ, ρ) := 4π

k∈R^∗\{0}

|k|⁻²ρ^∗_kρ_k.

We make the assumption thatV^ion is aR-periodic potential such that

∃m >3, C≥0 s.t. ∀k∈ R^∗, |V_k^ion| ≤C|k|^−m. (3.2)

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Note that this implies thatV^ion is inH^m−3/2−(Γ) for all >0, hence inC_#⁰(Γ) sincem−3/2− >3/2 for small enough. It is convenient to reformulate the TFW model in terms ofv=√

ρ. It can be easily seen that I^TFW= inf

E^TFW(v), v∈H_#¹(Γ),

Γ

|v|²=N

, (3.3)

where

E^TFW(v) =C_W 2

Γ

|∇v|²+C_TF

Γ

|v|^10/3+

Γ

V^ion|v|²+1

2D_Γ(|v|²,|v|²).

Let F(t) = C_TFt^5/3 and f(t) =F(t) = ⁵₃C_TFt^2/3. The function F is in C¹([0,+∞))∩C^∞((0,+∞)), is strictly convex on [0,+∞), and for all (t1, t₂)∈R+×R+,

|f(t²₂)t₂−f(t²₁)t₂−2f(t²₁)t²₁(t₂−t₁)| ≤ 70

27C_TFmax(t¹₁^/3, t¹₂^/3)|t2−t₁|². (3.4) The first and second derivatives ofE^TFW are respectively given by

E^TFW(v), w_H⁻¹

# ,H_#¹ = 2H^TFW_|v|² v, w, E^TFW(v)w₁, w_2H⁻¹

# ,H_#¹ = 2H^TFW_|v|² w₁, w₂+ 4D_Γ(vw₁, vw₂) + 4

Γ

f(|v|²)|v|²w₁w₂, where we have denoted byH^TFWρ the TFW Hamiltonian associated with the densityρ

H^TFWρ =−C_W

2 Δ +f(ρ) +V^ion+V_ρ^Coulomb, where

V_ρ^Coulomb(x) := 4π

k∈R^∗\{0}

|k|⁻²ρkek(x) (3.5)

is the R-periodic Coulomb potential generated by the R-periodic charge distribution ρ. Recall that V_ρ^Coulomb can also be defined as the unique solution inH_#¹(Γ) to

⎧⎪

⎪⎨

⎪⎪

⎩

−ΔVρ^Coulomb= 4π

ρ− |Γ|⁻¹

Γ

ρ ,

Γ

V_ρ^Coulomb= 0.

Let us recall (see [27] and the proof of Lemma 2 in [6]) that

• (3.1) has a unique minimizerρ⁰, and that the minimizers of (3.3) areuand−u, whereu= ρ⁰;

• uis inH_#^m+1/2−(Γ) for each >0 (hence inC_#²(Γ) sincem+ 1/2− >7/2 forsmall enough);

• u >0 onR³;

• usatisfies the Euler equation H^TFW_|u|² (u) =−C_W

2 Δu+ 5

3C_TFu^4/3+V^ion+V_u^Coulomb2

u=λu for some λ∈R, (the ground state eigenvalue of H^TFW_ρ0 , that is non-degenerate).

The planewave discretization of the TFW model is obtained by choosing:

(1) an energy cut-offE_c >0 or, equivalently, a finite dimensional Fourier space VN_c, the integerNc being related toE_c through the relationNc := [√

2E_cL/2π],

Rapide Note Highlight

(2) a cartesian gridGN_g with step sizeL/Ng whereNg∈N^∗ is such thatNg≥4Nc+ 1, and by considering the finite dimensional minimization problem

I_N^TFW_c,Ng = inf

E_N^TFW_g (vN_c), vN_c ∈VN_c,

Γ

|vN_c|²=N

, (3.6)

where

E_N^TFW_g (vN_c) = C_W 2

Γ

|∇vN_c|²+C_TF

Γ

IN_g(|vN_c|^10/3) +

Γ

IN_g(V^ion)|vN_c|² +1

2D_Γ(|vN_c|²,|vN_c|²),

INg denoting the interpolation operator introduced in the previous section. The Euler equation associated with (3.6) can be written as a nonlinear eigenvalue problem

∀vN_c ∈VN_c, H!^TFW_|uNc,Ng^,Ng_|2−λN_c,N_g

uN_c,N_g, vN_c

"

H_#⁻¹,H_#¹ = 0, where we have denoted by

H!ρ^TFW,Ng =−C_W

2 Δ +INg

5

3C_TFρ^2/3+V^ion

+V_ρ^Coulomb

the pseudospectral TFW Hamiltonian associated with the densityρ, and byλN_c,N_g the Lagrange multiplier of the constraint#

Γ|vNc|²=N. We therefore have

−C_W

2 ΔuN_c,N_g+ ΠN_c

$ IN_g

5

3C_TF|uN_c,N_g|^4/3+V^ion

+V_|^Coulomb_uNc,Ng|2

uN_c,N_g

%

=λN_c,N_guN_c,N_g. Under the condition thatNg≥4Nc+ 1, we have for allφ∈C_#⁰(Γ),

∀(k, l)∈ R^∗× R^∗ s.t. |k|,|l| ≤ 2π LNc,

Γ

IN_g(φ)e^∗kel=φ^FFTk−l, so that,H!^TFWu_Nc,Ng is defined onVNc by the Fourier matrix

&

H^TFW_|uNc,Ng^,Ng_|2

'

kl = C_W

2 |k|²δkl+5

3C_TF(

|uNc,Ng|^4/3)^FFT,Ng

k−l +(V^ion)^FFT

,N_g k−l

+ 4π (

|uNc,Ng|²)^FFT,Ng

k−l

|k−l|² (1−δkl),

where, by convention, the last term of the right hand side is equal to zero fork=l.

We also introduce the variational approximation of (3.3) I_N^TFW_c = inf

E^TFW(vN_c), vN_c ∈VN_c,

Γ

|vN_c|²=N

. (3.7)

Any minimizeruN_c to (3.7) satisfies the elliptic equation

−C_W

2 ΔuN_c+ ΠN_c

$5

3C_TF|uN_c|^4/3uN_c+V^ionuN_c+V_|^Coulomb_uNc|2 uN_c

%

=λN_cuN_c, (3.8) for someλN_c∈R.